Local Convexity Of The Tap Free Energy And Amp Convergence For Z2-Synchronization

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" summary

For Z2 synchronization at constant signal strength the result is that the TAP free energy has a single, locally strongly convex minimizer near the Bayes posterior moment for λ above the information threshold, with AMP shown to be an asymptotically stable fixed point and a hybrid spectral→AMP→natural gradient descent procedure provably converging for fixed instances when λ exceeds a larger λ0. Proof ingredients are Kac‑Rice calculus, conditional Gaussian comparison reducing the Hessian to a finite‑rank spiked GOE, state‑evolution for AMP and relative smoothness arguments for optimization.

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" chapters (7)

1. 0:00Start
2. 2:36What Is Z2 Synchronization
3. 6:12Variational Free Energy
4. 15:55The Iterative Algorithm
5. 21:20Proof
6. 22:47The Calculus Formula
7. 27:41Convergence of Emp

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